Two transformations are applied to the complex number $-3 - 8i$:

A $45^\circ$ rotation around the origin in the counter-clockwise direction.
A dilation, centered at the origin, with scale factor $\sqrt{2}.$

What is the resulting complex number?
Explanation: A $45^\circ$ rotation in the counter-clockwise direction corresponds to multiplication by $\operatorname{cis} 45^\circ = \frac{1}{\sqrt{2}} + \frac{i}{\sqrt{2}},$ and the dilation corresponds to multiplication by $\sqrt{2}.$  Therefore, both transformations correspond to multiplication by $\left( \frac{1}{\sqrt{2}} + \frac{i}{\sqrt{2}} \right) \sqrt{2} = 1 + i.$

[asy]
unitsize(0.5 cm);

pair A = (-3,-8), B = (5,-11);

draw((-4,0)--(6,0));
draw((0,-12)--(0,2));
draw((0,0)--A,dashed);
draw((0,0)--B,dashed);

dot("$-3 - 8i$", A, SW);
dot("$5 - 11i$", B, SE);
[/asy]

This means the image of $-3 - 8i$ is $(-3 - 8i)(1 + i) = \boxed{5 - 11i}.$